MinT - Best Known (57, 57+64, s)-Nets in Base 9

Best Known (57, 57+64, s)-Nets in Base 9

(57, 57+64, 114)-Net over F₉ — Constructive and digital

Digital (57, 121, 114)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (8, 40, 40)-net over F₉, using
  - net from sequence [i] based on digital (8, 39)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 8 and N(F) ≥ 40, using
      - a function field by SÃ©mirat [i]
2. digital (17, 81, 74)-net over F₉, using
  - net from sequence [i] based on digital (17, 73)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
      - a function field by GarcÃa and Quoos [i]

(57, 57+64, 184)-Net over F₉ — Digital

Digital (57, 121, 184)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(9¹²¹, 184, F₉, 3, 64) (dual of [(184, 3), 431, 65]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(9¹²¹, 186, F₉, 3, 64) (dual of [(186, 3), 437, 65]-NRT-code), using
  - constructionÂ X applied to AG(3;F,478P) âŠ‚ AG(3;F,486P) [i] based on
    1. linear OOA(9¹¹⁴, 181, F₉, 3, 64) (dual of [(181, 3), 429, 65]-NRT-code), using algebraic-geometric NRT-code AG(3;F,478P) [i] based on function field F/F₉ with g(F) = 50 and N(F) ≥ 182, using
      - a curve from the manYPoints database [i]
    2. linear OOA(9¹⁰⁶, 181, F₉, 3, 56) (dual of [(181, 3), 437, 57]-NRT-code), using algebraic-geometric NRT-code AG(3;F,486P) [i] based on function field F/F₉ with g(F) = 50 and N(F) ≥ 182 (see above)
    3. linear OOA(9⁷, 5, F₉, 3, 7) (dual of [(5, 3), 8, 8]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(9⁷, 9, F₉, 3, 7) (dual of [(9, 3), 20, 8]-NRT-code), using
        Reedâ€“Solomon NRT-code RS(3;20,9) [i]

(57, 57+64, 6467)-Net in Base 9 — Upper bound on s

There is no (57, 121, 6468)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 29 184122 644339 664506 580215 950672 364675 493286 249115 266063 868868 615065 834105 849139 792869 190248 478360 170554 365570 784257 > 9¹²¹ [i]